Download Sec 2.4 - 4 2.4 Further Applications of Linear Equations

Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 2.4 - 1
Chapter 2
Linear Equations and Applications
Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 2.4 - 2
2.4
Further Applications
of Linear Equations
Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 2.4 - 3
2.4 Further Applications of Linear Equations
Objectives
1. Solve problems about different
denominations of money.
2. Solve problems about uniform
motion.
3. Solve problems about angles.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 4
2.4 Further Applications of Linear Equations
Problems About Different Denominations of Money
Problem-Solving Hint
In problems involving money, use the fact that
number of monetary
units of the same kind
X denomination =
total monetary
value.
For example, 67 nickels have a monetary value of
$.05(67) = $3.35. Forty-two five dollar bills have a
value of $5(42) = $210.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 5
2.4 Further Applications of Linear Equations
Problems About Different Denominations of Money
Elise has been saving dimes and
quarters in a toy bank. Every time
she saves a coin, she pulls a small
lever, and the bank records the
number of coins that have been
deposited as well as the total
amount in the bank.
The bank contains $29.95, consisting
of 202 coins. How many of each
type coin does the bank contain?
Copyright © 2010 Pearson Education, Inc. All rights reserved.
202 coins
$29.95
Sec 2.4 - 6
2.4 Further Applications of Linear Equations
Problems About Different Denominations of Money
Continued.
Step 1
Read the problem.
The problem asks that we
find the number of dimes and
quarters that have been
saved in the bank.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
202 coins
$29.95
Sec 2.4 - 7
2.4 Further Applications of Linear Equations
Problems About Different Denominations of Money
Continued.
Step 2
Assign a variable.
Let x represent the number of dimes;
then 202 – x represents the number of quarters.
Denomination
$0.10
$0.25
Number of Coins
x
Total Value
0.10x
202 – x
0.25(202 – x)
202
$29.95
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 8
2.4 Further Applications of Linear Equations
Problems About Different Denominations of Money
Continued.
Step 3
Write an equation.
Denomination
$0.10
$0.25
Number of Coins
x
202 – x
202
From the last column of the table,
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Total Value
0.10x
0.25(202 – x)
$29.95
Totals
Sec 2.4 - 9
2.4 Further Applications of Linear Equations
Problems About Different Denominations of Money
Continued.
Step 4
Solve.
Multiply by 100.
Distributive prop.
Subtract 5,050.
Divide by –15.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 10
2.4 Further Applications of Linear Equations
Problems About Different Denominations of Money
Continued.
Step 5
State the answer.
Elise has 137 dimes and 202 – x = 202 – 137 = 65
quarters in the bank.
Caution
Always be sure your
Step 6
answer is reasonable!
Check.
The bank has 137 + 65 = 202 coins, and the value of
the coins is $.10(137) + $.25(65) = $29.95.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 11
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Opposite Directions)
W
E
Two snowmobiles leave the same place, one
going east and one going west. The
eastbound snowmobile averages 24 mph, and
the westbound snowmobile averages 32 mph.
How long will it take them to be 245 miles
apart?
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 12
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Opposite Directions)
Continued.
Step 1
Read the problem.
Caution
The sum of their distances
must be 245 mi. Each does
not travel 245 mi.
We must find the time it takes for the two
snowmobiles to be 245 miles apart.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 13
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Opposite Directions)
Continued.
32 mph
Step 2
Starting
point
24 mph
Total distance = 245 mph
Assign a variable.
The sketch shows what is happening in the problem.
Let x represent the time traveled by each snowmobile.
Rate
Time
Distance
Eastbound
24
x
24x
Westbound
32
x
32x
245
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 14
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Opposite Directions)
Continued.
Step 3 Write an equation.
Rate
Time
Distance
Eastbound
24
x
24x
Westbound
32
x
32x
245
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 15
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Opposite Directions)
Continued.
Step 4 Solve.
Combine like terms.
Divide by 56;
lowest terms
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 16
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Opposite Directions)
Continued.
Step 4 State the answer.
The snowmobiles travel
min.
hr, or 4 hr and 22½
Step 5 Check.
Distance traveled by eastbound
Distance traveled by westbound
Total distance traveled
Copyright © 2010 Pearson Education, Inc. All rights reserved.
OK
Sec 2.4 - 17
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Same Direction)
Brandon works 360 miles away from his home and
returns on weekends. For the trip home, he travels 6
hours on interstate highways and 1 hour on two-lane
roads. If he drives 25 mph faster on the interstate
highways than he does on the two-lane roads, determine
how fast he travels on each part of the trip home.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 18
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Same Direction)
Continued.
Step 2
Read the problem.
We are asked to find the speed Brandon
drives on the interstate highways and the
speed he drives on the two-lane roads.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 19
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Same Direction)
Continued.
Step 3 Assign a variable.
The problem asks for two speeds. We can let
Brandon’s speed on the two-lane highways be x. Then
the speed on the interstate highways must be x + 25.
For the interstate highways,
and for the two-lane roads
.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 20
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Same Direction)
Continued.
Step 2 Assign a variable.
Summarizing this information in a table, we have:
Rate
Time
Distance
Interstate
x + 25
6
6(x + 25)
Two-lane
x
1
x
360
Copyright © 2010 Pearson Education, Inc. All rights reserved.
 total
Sec 2.4 - 21
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Same Direction)
Continued.
Step 3 Write an equation.
Rate
Time
Distance
Interstate
x + 25
6
6(x + 25)
Two-lane
x
1
x
360
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 22
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Same Direction)
Continued.
Step 3 Solve.
Distributive prop.
Collect like terms.
Subtract 150.
Inverse prop.
Divide by 7.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 23
2.4 Further Applications of Linear Equations
Solving a Motion Problem (Same Directions)
Continued.
Step 5 State the answer.
Brandon drives the two-lane roads at a speed of 30
mph; he drives the interstate highways at x + 25 =
30 + 25 = 55 mph.
Step 6 Check by finding the distances using:
Distance traveled on interstate
highways
Distance traveled on two-lane
roads
Total distance traveled
Copyright © 2010 Pearson Education, Inc. All rights reserved.
360 mi
OK
Sec 2.4 - 24
2.4 Further Applications of Linear Equations
Solving Problems Involving Angles of a Triangle
From Euclidean Geometry
The sum of the angle measures of any triangle equal
180º.
C
A
E
B
Copyright © 2010 Pearson Education, Inc. All rights reserved.
D
F
Sec 2.4 - 25
2.4 Further Applications of Linear Equations
Finding Angle Measures
Find the value of x and determine the measure
of each angle in the figure.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 26
2.4 Further Applications of Linear Equations
Finding Angle Measures
Continued.
Step 1 Read the problem.
We are asked to find the measure of each
angle in the triangle.
Step 2 Assign a variable.
Let x represent the measure of one angle.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 27
2.4 Further Applications of Linear Equations
Finding Angle Measures
Continued.
Step 3
Write an equation.
The sum of the three measures shown in the figure
must equal 180º.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 28
2.4 Further Applications of Linear Equations
Finding Angle Measures
Continued.
Step 4
Solve the equation.
Collect like terms.
Subtract 110.
Divide by 2.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 29
2.4 Further Applications of Linear Equations
Finding Angle Measures
Continued.
Step 5
State the answer.
One angle measures 35º, another measures 2x + 25 =
2(35) + 25 = 95º, and the third angle measures 85 – x =
85 – 35 = 50º.
Step 6
Check.
Since 35º + 95º + 50º = 180º, the answer is correct.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Sec 2.4 - 30